The 47th problem of Euclid is so named because it was the 47th in a series of mathematical problems in his book of Mathematics. We perhaps know the problem better as the Pythagorean Theorem. In algebraic terms:

a² + b² = c²

where c is the hypotenuse while a and b are the sides of the triangle.

The typical example of this is the 3, 4 and 5 sided triangle.

The 47th problem of Euclid or Pythagorean Theorem is a common symbol to designate a Past Master. Sadly, far too many of them have no idea of its meaning in either its algebraic or symbolic forms.

• Mackey's Encyclopedia Article
• 1916 The Builder Article
• 1930 MSANA Short Talk Bulletin

FORTY-SEVENTH PROBLEM

The forty-seventh problem of Euclid's first book, which has been adopted as a symbol in the Master's Degree, is thus enunciated: "In any right-angled triangle, the square which is described upon the side subtending the right angle is equal to the squares described upon the sides which contain the right angle." Thus, in a triangle whose perpendicular is three feet, the square of which is nine, and whose base is four feet, the square of which is sixteen, the hypothenuse, or subtending side, will be five feet, the square of which will be twenty-five, which is the sum of nine and sixteen. This interesting problem, on account of its great utility in making calculations and drawing plans for buildings, is sometimes called the Carpenter's Theorem.

For the demonstration of this problem the world is indebted to Pythagoras, who, it is said, was so elated after making the discovery, that he made an offering of a hecatomb, or a sacrifice of a hundred oxen, to the gods. The devotion to learning which this religious act indicated in the mind of the ancient philosopher has induced Freemasons to adopt the problem as a memento, instructing them to be lovers of the arts and sciences.

The triangle, whose base is four parts, whose perpendicular is three, and whose hypothenuse is five, and which would exactly serve for a demonstration of this problem, was, according to Plutarch, a symbol frequently employed by the Egyptian priests, and hence it is called by M. Jomard, in his Exposition du Systeme Métrique des Amperes Egyptians, Exposition of the Ancient Egyptians System of Measurements, the Egyptian triangle. It was, with the Egyptians, the symbol of universal nature, the base representing Osiris, or the male principle; the perpendicular, Isis, or the female principle; and the hypothenuse, Horus, their son, or the produce of the two principles. They added that three was the first perfect odd number, that four was the square of two, the first even number, and that five was the result of three and two. But the Egyptians made a still more important use of this triangle. It was the standard of all their measures of extent, and was applied by them to the building of the pyramids. The researches of M. Jomard, on the Egyptian system of measures, published in the magnificent work of the French savants on Egypt, has placed us completely in possession of the uses made by the Egyptians of this forty-seventh problem of Euclid, and of the triangle which formed the diagram by which it was demonstrated.

If we inscribe within a circle a triangle, whose perpendicular shall be 300 parts, whose base shall be 500 parts, and whose hypotenuse shall be 500 parts, which, of course, bear the same proportion to each other as three, four, and five; then if we let a perpendicular fall from the angle of the perpendicular and base to the hypothenuse, and extend it through the hypothenuse to the circumference of the circle, this chord or lane will be equal to 480 parts, and the two segments of the hypothenuse, on each side of it, will be found equal, respectively, to 180 and 320. From the point where this chord intersects the hypothenuse let another lane fall perpendicularly to the shortest side of the triangle, and this line will be equal to 144 parts, while the shorter segment, formed by its junction with the perpendicular side of the triangle, will be equal to 108 parts. Hence, we may derive the following measures from the diagram: 500, 480, 400, 320, 180, 144, and 108, and all these without the slightest fraction. Supposing, then, the 400 to be cubits, we have the measure of the base of the great pyramid of Memphis. In the 400 cubits of the base of the triangle we have the exact length of the Egyptian stadium.

The 320 gives us the exact number of Egyptian cubits contained in the Hcbrew and Babylonian stadium. The stadium of Ptolemy is represented by the 480 cubits, or length of the line falling from the right angle to the circumference of the circle, through the hypothenuse. The number 180, which expresses the smaller segment of the hypothenuse being doubled, will give 360 cubits, which will be the stadum of Cleomedes. By doubling the 144, the result will be 288 cubits, or the length of the stadium of Archamedes; and by doublang the 108, we produce 216 cubits, or the precise value of the lesser Egyptian stadium.

Thus we get all the length measures used by the Egyptians; and since this triangle, whose sides are equal to three, four, and five, was the very one that most naturally would be used in demonstrating the forty-seventh problem of Euclid; and since by these three sides the Egyptians symbolized Osiris, Isis, and Horus, or the two producers and the product, the very principle, expressed in symbolic language, which constitutes the terms of the problem as enunciated by Pythagoras, that the sum of the squares of the two sides will produce the square of the third, we have no reason to doubt that the forty-seventh problem was well known to the Egyptian Priests, and by them communicated to Pythagoras.

Doctor Lardner, in his edition of Euclid, says:

Whether we consider the forty-seventh proposition with reference to the peculiar and beautiful relation established in it, or to its innumerable uses in every department of mathematical science, or to its fertility in the consequences derivable from it, it must certainly be esteemed the most celebrated and important in the whole of the elements, if not in the whole range, of mathematical science. It is by the influence of this proposition, and that which establishes the similitude of equiangular triangles, in the sixth book, that geometry has been brought under the dominion of algebra, and it is upon the same principles that the whole science of trigonometry is founded. The thirty-second and forty-seventh propositions are said to have been discovered by Pythagoras, and extraordinary accounts are given of his exultation upon his first perception of their truth. It is however, supposed by some that Pythagoras acquired a knowledge of them in Egypt, and was the first to make them known in Greece.

By Bro. C. C. Hunt, Grinnell, Iowa

The Master Mason will readily recognize this proposition as one of the emblems of the Third Degree. He will also recall the monitorial explanation of it there given, and possibly feel that it is an explanation which does not explain. He may not question the legendary history of it as given to him, but he does not understand why it should have been selected as a Masonic emblem, nor how it teaches Masons to be lovers of the arts and sciences. In fact there are many Masons who are not mathematicians and do not even know what the proposition is, and on this point the monitor is silent.

It is the object of this paper to briefly consider the history of the proposition and offer a few suggestions as to its Masonic significance. In doing this we may reach the conclusion that some of the monitorial statements are not historically true, or at least that they have not been proven. We will find, however, that the value of its symbolism does not depend on the truth of the historical statements given in the monitors, but is inherent in the proposition itself.

This will be hard for many Masons to understand. Through association of ideas, we are accustomed to think that the traditions which cluster around a central truth, are essential parts of that truth, and when critical investigation attacks the truth of the tradition, we feel it is an attack upon the truth itself. It is this trait of human nature which is the underlying cause of all religious persecution, and we are by no means free from it as Masons, though it is contrary to the fundamental principles of Masonry.

As members of the Masonic Research Society, it is our duty to search for the truth, no matter how much it may conflict with our preconceived notions or with traditions. If we but search aright, we will find that these traditions are but the outer garments with which time has clothed the truth, and that they are not its essential essence.

In our associations with each other we meet a kindred soul whom we learn to love and honor. We are told that he is the descendant of a great and honored name in history, and we say that the spirit of his forefathers has fallen upon him. Then some critic appears and shows that there is no proof of his illustrious ancestry, or perhaps entirely disproves it. What of it? Is he not the same friend we knew before? Has his soul lost any of its greatness? May not the spirit of a great soul have descended upon him, though his physical blood does not literally flow in his veins? We are told that the spirit of the prophet Elijah descended upon Elisha and centuries later appeared in John the Baptist. Yet there was no blood relationship between them. So it is with the proposition we are now studying. Its tradition and its history are both interesting, but its truth and the richness of its symbolism are not affected thereby.

In Euclid's Elements of Geometry there are thirteen books, and the subject we are considering is the forty seventh proposition of the first book. It is not a problem but a theorem, and is so called by Euclid. A problem in geometry is something to be done, as a figure to be drawn, while a theorem is something to be proved. This proposition is to prove, as Euclid states it, that "In any right-angled triangle, the square which is described on the side subtending (opposite) the right angle is equal to the square described on the sides which contain the right angle." The sides containing the right angle are called respectively the base and perpendicular, while the side opposite the right angle is called the hypotenuse.

Our monitors state that "This was the invention of our ancient friend and brother the great Pythagoras." This statement has been denied by many students of the subject. It has been claimed that this proposition was known to the Egyptians long before the time of Pythagoras, and that he learned it from them and carried it into Europe and Asia. We have no proof either for or against this claim. Pythagoras himself wrote nothing, and we know of his teachings only through the writings of his disciples. Vitruvius, a celebrated Roman architect of the time of Augustus Caesar, attributes the discovery of this proposition to Pythagoras. Plutarch quotes Apollydorus, a Greek painter of the 5th century B.C., as authority for the statement that Pythagoras sacrificed an ox on the discovery of this demonstration. Proclus credits Pythagoras with the first demonstration, but asserts that his proof was different from that given in Euclid. In fact so many writers, both ancient and modern, have attributed this proposition to Pythagoras that it is commonly called by his name, "The Theorem of Pythagoras."

On the other hand, the properties of the triangle whose sides are respectively, 3, 4, and 5, were certainly known to the Egyptians and were made the basis of all their measurement standards. We find evidence of this in their important buildings, many of them erected before the time of Pythagoras. We also find that this triangle was to them the symbol of universal nature.

May we not find an explanation of this apparent discrepancy in the statement of Plutarch that Pythagoras discovered the demonstration of the general proposition, but that the particular case in which the lengths of the sides are 3, 4, and 5, was earlier known to the Egyptians? Plutarch also thinks that the case in which the base and perpendicular are equal (as in the sides of a square) was likewise known to the Egyptians. This is called the classical form in Masonry and is the form usually found on the Master's carpet. Both these forms are rich in symbolism, and if known to the Egyptians, as they probably were, would naturally lead to the belief that the general demonstration was also known. Nevertheless it may be true, as claimed by so many writers, that to Pythagoras we owe the demonstration of the general proposition, which proved the theorem true for all possible cases. It was the delight of this philosopher to discover a universal principle underlying a concrete fact, and he must have attached a deeper meaning to the general truth than the Egyptians did to the special cases known to them. With him the science of numbers was the essence of all truth, and having discovered a proof for the general proposition, he set himself the task of finding right triangles whose sides can be expressed in numbers. Heron of Alexander and Proclus are authority for the statement that Pythagoras discovered the following method: Take any odd number for the shortest side; subtract one from the square of that number and divide the result by two; this will give the medium side; add one to the medium side and the result will be the hypotenuse or longest side. This is true as far as it goes, but it does not give all the right triangles which can be expressed in numbers.

The numerical symbolism of Pythagoras is an interesting study in itself and is closely allied to much of our Masonic symbolism, but that is outside the province of the present paper. It is simply mentioned here, because, while it is probably not true that he was raised to the sublime degree of a Master Mason as stated in our monitors, yet there is so much resemblance between his teachings and that of Masonry, that we can understand how the error might have occurred.

The monitor also states that Pythagoras celebrated his triumph in the discovery of this proposition by the sacrifice of a hecatomb (one hundred oxen). We can see how this may have been an outgrowth of the statement attributed to Apollodorus above. Ovid denies it and Hegel laughs at it, saying, "It was a feast of spiritual cognition, at the expense of the oxen." The strongest argument against it, however, is the fact that Pythagoras taught the doctrine of the transmigration of souls and forbade animal slaughter. However, when we consider that among many of the ancients the sacrifice of a number of oxen was their method of expressing their gratitude for a great triumph, we can understand how the tradition arose, and accept the fact of the joy without caring for the truth of the sacrifice.

Why should the discovery of this demonstration have been considered a great triumph? Because it is of the utmost importance to the science of geometry. Dionysius Lardner, in his edition of Euclid, quoted by Mackey, says, "Whether we consider the 47th problem with reference to the peculiar and beautiful relation established in it; or to its innumerable uses in every department of mathematical science, or to its fertility in the consequences derivable from it, it must certainly be esteemed the most celebrated and important in the whole of the elements, if not in the whole range of mathematical science. It is by the influence of this proposition and that which establishes the similitude of equiangular triangles (in the sixth book) that geometry has been brought under the dominion of algebra; and it is upon the same principle that the whole science of trigonometry is founded." The Encyclopedia Britannica calls it "One of the most important in the whole of geometry, and one which has been celebrated since the earliest times ;" and adds, "On this theorem almost all geometrical measurement depends, which cannot be directly obtained."

What is its significance in Masonry? Our monitors tell us that it teaches Masons to be lovers of the arts and sciences. Since it is so important a proposition in the science of mathematics, we can understand why it should be adopted as a symbol of scientific investigation, and to such an investigation all Masons are pledged in their search for truth, the great object of Masonic study. But has it not a deeper meaning? Dr. Lardner says it is the basis of the application of algebra to geometry. Algebra is the application of symbols to mathematics, and Masonry is the application of symbolism in character building. The Britannica says that mathematical measurements which cannot be directly obtained depend on this proposition. Yes, and as applied to Masonry, the highest truths of morality cannot be directly obtained. They must come to us indirectly through the medium, principally, of symbolism.

There is no apparent relation between the numbers 3, 4, and 5 and 5, 12, and 13, for instance; but when we raise these numbers from the first to the second power (that is, square them), we obtain 9, 16, and 25 in the first case, and 25, 144, and 169, in the second. In this form we notice in each case that the sum of the first two squares is equal to the third, and that the numbers in which we could at first see no relation are the sides of right angled triangles. So it is in life. Measured on the level of our lower natures, there is no relation between our own desires and our brother's needs. We are connected, it is true, as the sides of a triangle are connected, but there is no reason why we should not use him for the accomplishment of our own selfish purposes, irrespective of his welfare. It is only when we square our lives by the square of virtue, and our selfish desires are raised to spiritual purposes, that we perceive that our own welfare is intimately connected with that of our brother. His misfortunes are our misfortunes, and we can no more injure him and not be ourselves harmed thereby, than we can strike off our right hand and be none the worse by reason thereof.

We are traveling upon the level of time to our eternal destiny. We cannot stand still, but must constantly go forward. Shall we also go upward? All the time there is a spiritual force striving to lift us to higher levels. We may refuse to avail ourselves of it and remain in the depths of our lower nature; or we can accept it and allow its divine influence to shine in our lives. The base represents our earthly nature on the level of time; the perpendicular is the divine spirit striving to manifest itself through us. When these forces are squared to each other, their union becomes a constant onward and upward movement to the throne of God Himself. Pythagoras himself recognized this symbolism when he said that early in life he came to the place where two ways parted. One was easy and pleasant traveling; the other was rugged and tended upward. It necessitated hard climbing. Which was the way that led to life? All who travel there and find these two paths, know that he should choose the upward path, but the other seems so much more pleasant, and many are inclined to walk therein. They will try it a little while, and then return to the better way. But there is no turning back on the level of time. The farther they go on the lower level, the wider apart become the two ways, and the harder to cross from one to the other.

How often we have heard Masons say that there is no moral lesson to be derived from the 47th proposition of Euclid, and that it is not to be described as the symbol of any moral truth. Have they forgotten that there is not an observance or symbol of Masonry which has not a deep significance? Significance for what? Certainly as Masons it would have no especial significance for us unless it aided us in attaining the great purpose of our Order, "the uprearing of that spiritual temple, that house not made with hands, eternal in the heavens." It may well be that the significance is not recognized by us, but that by no means proves its nonexistence. It may be buried in the rubbish of preconceived opinions, and it only needs diligent digging to bring it to light.

We have here suggested but a few of the many applications of this symbol in the hope that it will stimulate others to more diligent research.

-Source: The Builder - February 1916

THE 47th PROBLEM

Containing more real food for thought, and impressing on the receptive mind a greater truth than any other of the emblems in the lecture of the Sublime Degree, the 47th problem of Euclid generally gets less attention, and certainly less than all the rest. Just why this grand exception should receive so little explanation in our lecture; just how it has happened, that, although the Fellowcraft’s degree makes so much of Geometry, Geometry’s right hand should be so cavalierly treated, is not for the present inquiry to settle. We all know that the single paragraph of our lecture devoted to Pythagoras and his work is passed over with no more emphasis than that given to the Bee Hive of the Book of Constitutions. More’s the pity; you may ask many a Mason to explain the 47th problem, or even the meaning of the word “hecatomb,” and receive only an evasive answer, or a frank “I don’t know - why don’t you ask the Deputy?” The Masonic legend of Euclid is very old - just how old we do not know, but it long antedates our present Master Mason’s Degree. The paragraph relating to Pythagoras in our lecture we take wholly from Thomas Smith Webb, whose first Monitor appeared at the close of the eighteenth century.

It is repeated here to refresh the memory of those many brethren who usually leave before the lecture:

“The 47th problem of Euclid was an invention of our ancient friend and brother, the great Pythagoras, who, in his travels through Asia, Africa and Europe was initiated into several orders of Priesthood, and was also Raised to the Sublime Degree of Master Mason. This wise philosopher enriched his mind abundantly in a general knowledge of things, and more especially in Geometry. On this subject he drew out many problems and theorems, and, among the most distinguished, he erected this, when, in the joy of his heart, he exclaimed Eureka, in the Greek Language signifying “I have found it,” and upon the discovery of which he is said to have sacrificed a hecatomb. It teaches Masons to be general lovers of the arts and sciences.” Some of facts here stated are historically true; those which are only fanciful at least bear out the symbolism of the conception. In the sense that Pythagoras was a learned man, a leader, a teacher, a founder of a school, a wise man who saw God in nature and in number; and he was a “friend and brother.” That he was “initiated into several orders of Priesthood” is a matter of history. That he was “Raised to the Sublime Degree of Master Mason” is of course poetic license and an impossibility, as the “Sublime Degree” as we know it is only a few hundred years old - not more than three at the very outside. Pythagoras is known to have traveled, but the probabilities are that his wanderings were confined to the countries bordering the Mediterranean. He did go to Egypt, but it is at least problematical that he got much further into Asia than Asia Minor. He did indeed “enrich his mind abundantly” in many matters, and particularly in mathematics. That he was the first to “erect” the 47th problem is possible, but not proved; at least he worked with it so much that it is sometimes called “The Pythagorean problem.” If he did discover it he might have exclaimed “Eureka” but the he sacrificed a hecatomb - a hundred head of cattle - is entirely out of character, since the Pythagoreans were vegetarians and reverenced all animal life.

Pythagoras was probably born on the island of Samos, and from contemporary Grecian accounts was a studious lad whose manhood was spent in the emphasis of mind as opposed to the body, although he was trained as an athlete. He was antipathetic to the licentiousness of the aristocratic life of his time and he and his followers were persecuted by those who did not understand them. Aristotle wrote of him: “The Pythagoreans first applied themselves to mathematics, a science which they improved; and penetrated with it, they fancied that the principles of mathematics were the principles of all things.”

It was written by Eudemus that: “Pythagoreans changed geometry into the form of a liberal science, regarding its principles in a purely abstract manner and investigated its theorems from the immaterial and intellectual point of view,” a statement which rings with familiar music in the ears of Masons.

Diogenes said “It was Pythagoras who carried Geometry to perfection,” also “He discovered the numerical relations of the musical scale.” Proclus states: “The word Mathematics originated with the Pythagoreans!”

The sacrifice of the hecatomb apparently rests on a statement of Plutarch, who probably took it from Apollodorus, that “Pythagoras sacrificed an ox on finding a geometrical diagram.” As the Pythagoreans originated the doctrine of Metempsychosis which predicates that all souls live first in animals and then in man - the same doctrine of reincarnation held so generally in the East from whence Pythagoras might have heard it - the philosopher and his followers were vegetarians and reverenced all animal life, so the “sacrifice” is probably mythical. Certainly there is nothing in contemporary accounts of Pythagoras to lead us to think that he was either sufficiently wealthy, or silly enough to slaughter a hundred valuable cattle to express his delight at learning to prove what was later to be the 47th problem of Euclid.

In Pythagoras’ day (582 B.C.) of course the “47th problem” was not called that. It remained for Euclid, of Alexandria, several hundred years later, to write his books of Geometry, of which the 47th and 48th problems form the end of the first book. It is generally conceded either that Pythagoras did indeed discover the Pythagorean problem, or that it was known prior to his time, and used by him; and that Euclid, recording in writing the science of Geometry as it was known then, merely availed himself of the mathematical knowledge of his era.

It is probably the most extraordinary of all scientific matters that the books of Euclid, written three hundred years or more before the Christian era, should still be used in schools. While a hundred different geometries have been invented or discovered since his day, Euclid’s “Elements” are still the foundation of that science which is the first step beyond the common mathematics of every day. In spite of the emphasis placed upon geometry in our Fellowcrafts degree our insistence that it is of a divine and moral nature, and that by its study we are enabled not only to prove the wonderful properties of nature but to demonstrate the more important truths of morality, it is common knowledge that most men know nothing of the science which they studied - and most despised - in their school days. If one man in ten in any lodge can demonstrate the 47th problem of Euclid, the lodge is above the common run in educational standards!

And yet the 47th problem is at the root not only of geometry, but of most applied mathematics; certainly, of all which are essential in engineering, in astronomy, in surveying, and in that wide expanse of problems concerned with finding one unknown from two known factors. At the close of the first book Euclid states the 47th problem - and its correlative 48th - as follows:

“47th - In every right angle triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.” “48th - If the square described of one of the sides of a triangle be equal to the squares described of the other two sides, then the angle contained by these two is a right angle.”

This sounds more complicated than it is. Of all people, Masons should know what a square is! As our ritual teaches us, a square is a right angle or the fourth part of a circle, or an angle of ninety degrees. For the benefit of those who have forgotten their school days, the “hypotenuse” is the line which makes a right angle (a square) into a triangle, by connecting the ends of the two lines which from the right angle.

For illustrative purposes let us consider that the familiar Masonic square has one arm six inches long and one arm eight inches long. If a square be erected on the six inch arm, that square will contain square inches to the number of six times six, or thirty-six square inches. The square erected on the eight inch arm will contain square inches to the number of eight times eight, or sixty-four square inches.

The sum of sixty-four and thirty-six square inches is one hundred square inches.

According to the 47th problem the square which can be erected upon the hypotenuse, or line adjoining the six and eight inch arms of the square should contain one hundred square inches. The only square which can contain one hundred square inches has ten inch sides, since ten, and no other number, is the square root of one hundred. This is provable mathematically, but it is also demonstrable with an actual square. The curious only need lay off a line six inches long, at right angles to a line eight inches long; connect the free ends by a line (the Hypotenuse) and measure the length of that line to be convinced - it is, indeed, ten inches long.

This simple matter then, is the famous 47th problem.

But while it is simple in conception it is complicated with innumerable ramifications in use.

It is the root of all geometry. It is behind the discovery of every unknown from two known factors. It is the very cornerstone of mathematics.

The engineer who tunnels from either side through a mountain uses it to get his two shafts to meet in the center.

The surveyor who wants to know how high a mountain may be ascertains the answer through the 47th problem.

The astronomer who calculates the distance of the sun, the moon, the planets and who fixes “the duration of time and seasons, years and cycles,” depends upon the 47th problem for his results.

The navigator traveling the trackless seas uses the 47th problem in determining his latitude, his longitude and his true time.

Eclipses are predicated, tides are specified as to height and time of occurrence, land is surveyed, roads run, shafts dug, and bridges built because of the 47th problem of Euclid - probably discovered by Pythagoras - shows the way.

It is difficult to show “why” it is true; easy to demonstrate that it is true. If you ask why the reason for its truth is difficult to demonstrate, let us reduce the search for “why” to a fundamental and ask “why” is two added to two always four, and never five or three?” We answer “because we call the product of two added to two by the name of four.” If we express the conception of “fourness” by some other name, then two plus two would be that other name. But the truth would be the same, regardless of the name.

So it is with the 47th problem of Euclid. The sum of the squares of the sides of any right angled triangle - no matter what their dimensions - always exactly equals the square of the line connecting their ends (the hypotenuse). One line may be a few 10’s of an inch long - the other several miles long; the problem invariably works out, both by actual measurement upon the earth, and by mathematical demonstration.

It is impossible for us to conceive of a place in the universe where two added to two produces five, and not four (in our language). We cannot conceive of a world, no matter how far distant among the stars, where the 47th problem is not true. For “true” means absolute - not dependent upon time, or space, or place, or world or even universe. Truth, we are taught, is a divine attribute and as such is coincident with Divinity, omnipresent.

It is in this sense that the 47th problem “teaches Masons to be general lovers of the art and sciences.” The universality of this strange and important mathematical principle must impress the thoughtful with the immutability of the laws of nature. The third of the movable jewels of the entered Apprentice Degree reminds us that “so should we, both operative and speculative, endeavor to erect our spiritual building (house) in accordance with the rules laid down by the Supreme Architect of the Universe, in the great books of nature and revelation, which are our spiritual, moral and Masonic Trestleboard.”

Greatest among “the rules laid down by the Supreme Architect of the Universe,” in His great book of nature, is this of the 47th problem; this rule that, given a right angle triangle, we may find the length of any side if we know the other two; or, given the squares of all three, we may learn whether the angle is a “Right” angle, or not. With the 47th problem man reaches out into the universe and produces the science of astronomy. With it he measures the most infinite of distances. With it he describes the whole framework and handiwork of nature. With it he calcu-lates the orbits and the positions of those “numberless worlds about us.” With it he reduces the chaos of ignorance to the law and order of intelligent appreciation of the cosmos. With it he instructs his fellow-Masons that “God is always geometrizing” and that the “great book of Nature” is to be read through a square.

Considered thus, the “invention of our ancient friend and brother, the great Pythagoras,” becomes one of the most impressive, as it is one of the most important, of the emblems of all Freemasonry, since to the initiate it is a symbol of the power, the wisdom and the goodness of the Great Articifer of the Universe. It is the plainer for its mystery - the more mysterious because it is so easy to comprehend.

Not for nothing does the Fellowcraft’s degree beg our attention to the study of the seven liberal arts and sciences, especially the science of geometry, or Masonry. Here, in the Third Degree, is the very heart of Geometry, and a close and vital connection between it and the greatest of all Freemasonry’s teachings - the knowledge of the “All-Seeing Eye.”

He that hath ears to hear - let him hear - and he that hath eyes to see - let him look! When he has both listened and looked, and understood the truth behind the 47th problem he will see a new meaning to the reception of a Fellowcraft, understand better that a square teaches morality and comprehend why the “angle of 90 degrees, or the fourth part of a circle” is dedicated to the Master!